Browsing by Subject "Degree of freedom"
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Item A four moment solution to the Behrens-Fisher problem(Texas Tech University, 1981-12) Scariano, Stephen MarkNot availableItem Development of a non-Gaussian closure scheme for dynamic systems involving non-linear inertia(Texas Tech University, 1985-12) Soundararajan, AravamudhanThe main objective of this investigation is to develop a non-Gaussian closure scheme adapted for the analysis of random response statistics of nonlinear dynamic systems which are subjected to parametric random excitations. The scheme is based on the asymptotic expansion of the non-Gaussian probability density. The technique is found to have two main advantages. The first is that it resolves an observed contradiction of results obtained by other techniques. The second is that it explores new response characteristics not predicted by other methods. The scheme is applied to a number of dynamic systems possessing single and two degrees-of-freedom and various types of nonlinearities. Numerical solutions are obtained and their validity is examined according to certain criteria based on the preservation of moment properties and Schwarz's inequality. Higher order terms are considered to examine the convergence of the Edgeworth expansion. It is found that the inclusion of the third and fourth order semi-invariants is adequate for the series convergence. The response of nonlinear single degree-of-freedom systems exhibits the occurrence of a jump in the response statistics at a certain excitation level which is mainly governed by the system linear damping factor. This new feature may be attributed to the fact that the non-Gaussian closure more adequately models the nonlinearity, and thus results in characteristics that are similar to those of deterministic nonlinear systems. The method is also used to determine the random response of an elevated water tower subjected to a random ground motion. The tower is represented by a two-degree-of-freedom system with cubic nonlinear coupling. In the neighborhood of the internal resonance condition r= w2/w1 =1/3, where w1 and w2 are the normal mode frequencies of the system, the nonlinear modal interaction takes the form of energy exchange between the two modes. Unlike the Gaussian solution, the non-Gaussian closure solution is found to achieve a strictly stationary response in the time domain. The response mean squares are presented as functions of the internal resonance detuning parameter r = 1/3 + 0(e), where £ is a small parameter, for various system parameters. Unbounded response mean squares are found to take place at regions above and below certain values of the internal resonance r=l/3. For regions well remote from the exact internal tuning the system exhibits the features of the linear response.Item Experimental investigation of stochastic vibration of nonlinear structures(Texas Tech University, 1986-12) Sullivan, Douglas GrantThe purpose of this experimental investigation is to measure the dynamic response of an aeroelastic structure involving nonlinear coupling to random parametric vibration. Two main series of tests are conducted under two different bandwidths. The first test corresponds to isolation of the first normal mode natural frequency. The second test corresponds to a bandwidth which covers the second normal mode frequency. The tests are conducted when the structure is tuned internally such that the second normal mode frequency is twice the first normal mode frequency. Experimental measurements are processed to estimate the response mean squares. The influence of excitation spectral density level and internal detuning on the response mean squares is examined. The results confirm regions of instability as predicted in harmonic excitation but there is no evidence of the well known saturation phenomenon. The results differ from the theoretical results obtained for the same model under white band random excitation. This disagreement is mainly due to the fact that the excitation is represented by a physical white noise process in the analytical model while it is a band— limited process in the actual experiment.Item Linear and autoparametric modal analysis of aeroelastic structural systems(Texas Tech University, 1984-05) Woodall, Tommy DaleThis investigation deals with the linear modal analysis and autoparametric interaction of aeroelastic systems such as an airplane fuselage and wing with fuel storage. The mathematical modeling is derived by applying Lagrange's equations taking into consideration the Christoffel symbol of the first kind to account for the nonlinear coupling of the system coordinates, velocities, and accelerations. The linear modal analysis will be obtained by considering the linear, conservative portion of the equations of motion. The normal mode frequencies and the associated mode shapes are obtained in terms of the system parameters. The main objective of the linear analysis is to explore the critical regions of autoparametric (or internal) resonance conditions, £kiwi=0 (where ki are integers and wi are the normal mode frequencies). The results show that for certain system parameters the condition of internal resonance is satisfied. The dynamic behavior of the structure in the neighborhood of internal resonance conditions is obtained by considering the nonlinear coupling of the normal modes. The asymptotic approximation technique due to Struble is employed. Three groups of internal and normal resonance conditions are obtained from the secular terms of the first-order perturbational equations. The transient and steady-state responses cure obtained numerically by using the IBM Continuous System Modeling Program (CSMP) with double precision Milne integration. The transient response shows a build up in the interacted modes to a level which exceeds the steady-state response. In addition, the excited mode is suppressed by virtue of the nonlinear feedback of other modes. Under certain conditions, the steady-state response is derived analytically. It is concluded that the nonlinear modal analysis reveals certain types of response characteristics which cannot be interpreted within the framework of the linear theory of small oscillations.Item Random modal interaction of a non-linear aeroelastic structure(Texas Tech University, 1985-08) Hedayati, ZhianThe modal interaction of an aeroelastic structure subjected to random wide band excitation is investigated. An analytical model, represented by three degrees of freedom, is adopted. The equations of motion are derived by employing Lagrange's equation. The Fokker-Planck equation approach is used to generate the statistical dynamic moment equations of the response. Linear and non-linear modal interactions are obtained for various system parameters. The linear modal analysis involves the determination of the normal mode eigenvalues in terms of the system parameters. The main objective of this standard analysis is to define the critical regions of internal resonance of the sum type ω3 = ω1 + ω2, where o)i are the system eigenvalues. Analytical solutions are obtained for the mean square response of the linearized system with constant and randomly varying stiffnesses. The results provide mean square stability criteria for the linear case with random stiffness. The linear response statistics are used as a reference to measure the departure of the non-linear system response in the neighborhood of the internal resonance condition. For the non-linear case the differential equations of the response moments are found to form an infinite coupled set which is closed via a cumulant-neglect scheme. The resulting closed first and second order moment equations, 27 in total, are solved by numerical integration. The response at the critical internal resonance demonstrates a deviation from the corresponding linear response. The results indicate that the autoparametric interaction takes place in the form of energy transfer between the three modes, and large amplitude motion of one mode associated with suppression of the other two modes.Item Response of a nonlinear two-degree-of-freedom system to a horizontal harmonic excitation(Texas Tech University, 1985-12) Li, WenlungAn elastic structure containing a fluid subjected to a horizontal sinusoidal excitation is investigated. The system is found to include cubic nonlinearities. The system response is determined by using the multiple scales asymptotic approximation method. The method predicts that primary resonances may occur when the excitation frequency, Ω is close to either the first mode natural frequency, ω1, or the second mode natural frequency, ω2. The system behavior under the fourth order internal resonance condition (ω2 ≈ 3ω1) is predicted. The system response under conditions of primary resonances (Ω ≈ω1 and Ω≈ω2), together with internal resonance is also considered. Other features, such as amplitude jump phenomenon and chaotic-like response have been observed. Two possible responses have been found when Ω is near ω2 = unlmodal response and autoparametric interaction response. The boundaries of these two motions are defined in the excitation amplitude - frequency plane. Moreover, the so called "static attractor" is also observed.